Converting Between Deferred Immediate and Due Values

Why Payment Dates Matter in a Conversion

A deferred annuity-immediate and a deferred annuity-due can have the same number of payments and the same deferral period, yet different present values. Under the usual convention, the due payments occur one period earlier than the immediate payments. At a positive effective interest rate, that timing difference makes the due value larger.

The conversion is straightforward only after the payment dates are clear. This tutorial uses a constant effective rate per payment period, level payments, and a present value measured at time 0. The examples use annual payments, so $i$ is the effective annual rate. There are no expenses, taxes, or contingencies affecting the payments.

Definition: A deferred annuity-immediate with $n$ level payments of amount $R$, deferred $k$ complete periods, pays at times $k+1,k+2,\ldots,k+n$. Its time-0 present value is $R v^k a_{\overline{n}|}$. A deferred annuity-due with the same $n$, $R$, and $k$ pays at times $k,k+1,\ldots,k+n-1$. Its time-0 present value is $R v^k \ddot{a}_{\overline{n}|}$. Here $v=1/(1+i)$, and $k$ is a nonnegative integer.

In this convention, a deferral of $k$ periods for a due annuity means the first due payment is at time $k$. For an immediate annuity deferred $k$ periods, the first payment is one period later, at time $k+1$. The term “same deferral” therefore does not mean that the two streams begin on the same date.

The Conversion Relationship

For equal payment amounts, equal numbers of payments, and equal deferral periods, the deferred due value is $1+i$ times the deferred immediate value. The reason is that each due payment occurs exactly one period earlier than its corresponding immediate payment.

\begin{aligned} PV_{\text{immediate}} &=R\left(v^{k+1}+v^{k+2}+\cdots+v^{k+n}\right)\\ &=R v^k a_{\overline{n}|},\\[4pt] PV_{\text{due}} &=R\left(v^k+v^{k+1}+\cdots+v^{k+n-1}\right)\\ &=R v^k \ddot{a}_{\overline{n}|}. \end{aligned}

Each discount factor in the due sum is $1+i$ times the corresponding factor in the immediate sum: for example, $v^{k+j-1}=(1+i)v^{k+j}$, since $v=1/(1+i)$. Applying this to all $n$ corresponding payments gives the conversion.

Theorem: If the two deferred annuities have the same payment amount, payment frequency, number of payments, and deferral $k$, then $$ PV_{\text{due}}=(1+i)PV_{\text{immediate}}. $$ Equivalently, $PV_{\text{immediate}}=vPV_{\text{due}}$. This relationship assumes the due payment dates are exactly one period earlier than the corresponding immediate payment dates. If the streams do not have that timing relationship, do not apply the conversion automatically.

The same result follows from the standard annuity-factor identity $\ddot{a}_{\overline{n}|}=(1+i)a_{\overline{n}|}$. The deferral factor $v^k$ appears in both values, so it does not change their ratio. This cancellation is why the conversion factor depends on the interest rate per payment period, not on the length of the deferral.

Write the payment dates.
For a deferred immediate annuity with deferral $k$, list $k+1$ through $k+n$. For a deferred due annuity with deferral $k$, list $k$ through $k+n-1$.

Compare the streams.
Check whether payment amounts, payment counts, and deferrals match, so that every due payment is one period earlier than its immediate counterpart.

Apply the conversion.
For matching streams, multiply the immediate value by $1+i$ to obtain the due value, or multiply the due value by $v$ to obtain the immediate value.

Check the dates if anything differs.
If the first payment dates do not have the required one-period offset, use the actual dates and discount each payment accordingly.

Worked Conversions

Worked Example: Converting Equal Deferred Streams

A hypothetical fund will make three annual payments of $1,200. The effective annual interest rate is 25%, and the deferral is two years. Find the time-0 values of the deferred immediate and deferred due versions, then verify their relationship from the payment dates.

Method and timing. For the immediate version, the payments occur at times 3, 4, and 5. For the due version with the same two-year deferral, they occur at times 2, 3, and 4. The due payments are each one year earlier.

At $i=0.25$, the discount factor is $v=1/1.25=0.8$. The immediate annuity factor is

a_{\overline{3}|}=v+v^2+v^3=0.8+0.64+0.512=1.952.

The deferred immediate value is

PV_{\text{immediate}} =1{,}200(0.8)^2(1.952) =1{,}200(0.64)(1.952) =768(1.952) =\$1{,}499.136.

For the due version,

\ddot{a}_{\overline{3}|}=1+v+v^2=1+0.8+0.64=2.44,

PV_{\text{due}} =1{,}200(0.64)(2.44) =768(2.44) =\$1{,}873.92.

Check by discounting payments. The immediate value is $1{,}200(0.512+0.4096+0.32768)=1{,}200(1.24928)=\$1{,}499.136$. The due value is $1{,}200(0.64+0.512+0.4096)=1{,}200(1.5616)=\$1{,}873.92$. Also, $1.25(\$1{,}499.136)=\$1{,}873.92$, as required. Rounded to the nearest cent, the values are $1,499.14 and $1,873.92. The common mistake this check avoids is treating the two streams as if they began on the same date.

Worked Example: Finding an Equivalent Due Payment

A hypothetical deferred annuity-immediate pays $1,250 annually for three years, with a two-year deferral. The effective annual rate is 25%. Find the level annual payment for a deferred annuity-due with the same deferral and number of payments that has the same time-0 present value.

Method. With the same deferral and term, the due value for a given payment amount is $1+i=1.25$ times the immediate value. Therefore, to keep the value unchanged, the due payment must be the immediate payment divided by 1.25:

R_{\text{due}}=\frac{\$1{,}250}{1.25}=\$1{,}000.

Verify using the payment dates. The immediate payments occur at times 3, 4, and 5. Their present value is

\$1{,}250(0.512+0.4096+0.32768) =\$1{,}250(1.24928) =\$1{,}561.60.

The due payments occur at times 2, 3, and 4. Their present value is

\$1{,}000(0.64+0.512+0.4096) =\$1{,}000(1.5616) =\$1{,}561.60.

The two present values agree. The common mistake avoided here is multiplying the payment amount by $1+i$; that factor converts the value of an unchanged stream, not the payment needed to preserve its value.

Worked Example: Same First Payment Date, Different Deferrals

A hypothetical immediate annuity pays $500 at times 3, 4, and 5. A due annuity also pays $500 at times 3, 4, and 5. The effective annual rate is 25%. Find their time-0 present values and explain why multiplying one value by 1.25 would be incorrect.

The immediate annuity has deferral $k=2$, since its payments begin at time $k+1=3$. The due annuity has deferral $k=3$, since its payments begin at time $k=3$. Their payment dates are identical, so their present values must be identical:

PV=\$500(0.512+0.4096+0.32768) =\$500(1.24928) =\$624.64.

If we used the same-deferral conversion, we might incorrectly multiply $624.64 by 1.25 and report $780.80. That conversion does not apply: it compares a due stream with deferral 2, paying at times 2, 3, and 4, against an immediate stream with deferral 2, paying at times 3, 4, and 5. In this example, the two streams already have the same payment dates. The direct discounted sum confirms the correct value of $624.64 for each.

When the Conversion Does Not Apply

The factor $1+i$ is not a general adjustment for any pair of deferred annuities. It applies when the streams correspond payment by payment, with each due payment exactly one period before the immediate payment of the same amount. Check the stream details before using it.

Comparison	Can use $PV_{\text{due}}=(1+i)PV_{\text{immediate}}$?	Reason
Same $k$, same $n$, same level payment, same period length	Yes	The due dates are one period earlier.
Same first payment date and same payment dates	No, not as a conversion	The values are equal for equal payments because the dates are the same.
Different payment counts or payment amounts	Not automatically	The streams may not match payment by payment.
Different payment frequencies or rate periods	Not with the same $i$ without adjustment	The conversion uses the effective rate for one payment period.

A reliable check is to write the payment dates explicitly. If every corresponding payment in the due stream is one period earlier, its discount factor is $1+i$ times the factor for that immediate payment. If the dates do not satisfy this condition, calculate from the actual dates rather than forcing a conversion.

This distinction matters in deferred-annuity problems because the word “deferred” can describe either the length of a wait or the date of the first payment. State the convention being used. The formulas here define deferral $k$ by the standard payment dates shown above; if a problem instead specifies a first payment date directly, use that date to identify the cash flows.

Key takeaway: With equal payment amounts, equal payment counts, and equal deferral $k$, a deferred annuity-due pays one period earlier than a deferred annuity-immediate, so its value is $1+i$ times as large. When deferrals differ, compare payment dates before converting.

Check Your Understanding

For each problem, identify the timing before applying any conversion.

At an effective annual rate of 10%, find the time-0 values of a four-payment deferred annuity-immediate paying $3,000 per year with $k=2$, and the corresponding deferred annuity-due with the same payment, term, and deferral. Round to the nearest cent. (Answer: immediate $7,859.17; due $8,645.09.)
At an effective annual rate of 25%, a three-payment deferred immediate annuity with $k=1$ pays $1,250 annually. Find the annual payment for a deferred due annuity with the same $k$ and term that has equal present value, and state that common value. (Answer: due payment $1,000; common present value $1,952.00.)
At an effective annual rate of 20%, a four-payment deferred due annuity with $k=2$ has a present value of $4,392. Find the present value of the corresponding deferred immediate annuity with the same payment, term, and deferral. (Answer: $3,660.00.)
At an effective annual rate of 10%, two equal-payment annuities each pay $600 at times 5, 6, and 7. One is described as immediate and the other as due. Find each time-0 value, rounded to the nearest cent, and explain why their values are equal. (Answer: $1,019.13 each; their payment dates are identical.)
At an effective annual rate of 25%, a deferred immediate annuity with $k=2$ pays $500 for three years. A deferred due annuity has $k=2$, the same term, and pays $400 per year. Determine whether their present values are equal and give the value. (Answer: Yes; each is $624.64.)

Comparison	Can use \(PV_{\text{due}}=(1+i)PV_{\text{immediate}}\)?	Reason
Same \(k\), same \(n\), same level payment, same period length	Yes	The due dates are one period earlier.
Same first payment date and same payment dates	No, not as a conversion	The values are equal for equal payments because the dates are the same.
Different payment counts or payment amounts	Not automatically	The streams may not match payment by payment.
Different payment frequencies or rate periods	Not with the same \(i\) without adjustment	The conversion uses the effective rate for one payment period.

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